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Refined curve counting with tropical geometry

Part of: Projective and enumerative geometry

Published online by Cambridge University Press:  18 August 2015

Florian Block  and
Lothar Göttsche
Florian Block
Affiliation:
Department of Mathematics, University of California, Berkeley, CA, USA email florian.s.block@gmail.com
Lothar Göttsche
Affiliation:
International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy email gottsche@ictp.it
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Abstract

The Severi degree is the degree of the Severi variety parametrizing plane curves of degree $d$ with ${\it\delta}$ nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable $y$, which are conjecturally equal, for large $d$. At $y=1$, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a tropical description of the refined Severi degrees, in terms of a refined tropical curve count for all toric surfaces. We also refine the equivalent count of floor diagrams for Hirzebruch and rational ruled surfaces. Our description implies that, for fixed ${\it\delta}$, the refined Severi degrees are polynomials in $d$ and $y$, for large $d$. As a consequence, we show that, for ${\it\delta}\leqslant 10$ and all $d\geqslant {\it\delta}/2+1$, both refinements of Göttsche and Shende agree and equal our refined counts of tropical curves and floor diagrams.

Keywords

Severi varietyrefined Severi degreeGöttsche conjectureWelschinger invarianttropical geometryfloor diagram

MSC classification

Primary: 14N10: Enumerative problems (combinatorial problems)
Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 14T05: Tropical geometry
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 1 , January 2016 , pp. 115 - 151
Copyright
© The Authors 2015 

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